HMC Parameters

HMC Parameters

Tuning the Hybrid Monte Carlo (HMC) updating scheme in ALF. HMC is available for models with continuous auxiliary fields (OP_V%type=3).

Reference: The algorithm is derived in the documentation (PDF), Section on Hybrid Monte Carlo.

Parameters

All HMC parameters are set in the &VAR_QMC namelist in the parameters file.

Parameter	Default	Type	Description
HMC	.false.	Logical	Enable HMC updates
Delta_t_Langevin_HMC	0.0	Real	Leap-frog integration step size $\delta t_m$
Leapfrog_Steps	0	Integer	Number of leap-frog steps $N_\text{LF}$ per trajectory
N_HMC_sweeps	1	Integer	Number of HMC trajectories between sequential sweeps
Sequential	.true.	Logical	Keep sequential single-site updates active alongside HMC

The trajectory length is $T_m$ = Delta_t_Langevin_HMC × Leapfrog_Steps.

Example

&VAR_QMC
HMC                    = .true.
Sequential             = .true.
Delta_t_Langevin_HMC   = 0.1d0
Leapfrog_Steps         = 10
N_HMC_sweeps           = 5
NSweep                 = 100
NBin                   = 20
Nwrap                  = 10
/

Guidelines

Step Size and Acceptance Rate

The leap-frog integrator introduces an $\mathcal{O}(\delta t_m^2)$ discretization error in the Hamiltonian. The Metropolis accept/reject step corrects for this, but large errors lead to low acceptance.

• Target acceptance rate: ~70–80%.
• Check the acceptance rate in the info file after a run (Acceptance_HMC).
• If acceptance is too low: reduce Delta_t_Langevin_HMC.
• If acceptance is very high (>95%): you're being too conservative — increase Delta_t_Langevin_HMC or Leapfrog_Steps to explore phase space faster.

Trajectory Length

• The trajectory length $T_m$ controls how far each HMC proposal moves in configuration space.
• Too short: proposals are close to the current state (inefficient, like random walk).
• Too long: computational cost per trajectory is high without proportional gain.
• A reasonable starting point is $T_m \approx 1$.

Combining with Sequential Updates

Always keep Sequential = .true. when using HMC. The documentation explicitly warns:

"In order to address potential ergodicity issues it is crucial to integrate both sequential and hybrid molecular dynamics moves, rather than relying solely on hybrid molecular dynamics."

N_HMC_sweeps controls how many HMC trajectories are performed between sequential sweeps. Increasing this value shifts the balance toward HMC moves.

Mass Matrix (Apply_B_HMC)

The HMC Hamiltonian is:

$$H(\mathbf{p}, \mathbf{s}) = \frac{1}{2} \mathbf{p}^T M^{-1} \mathbf{p} + S(\mathbf{s})$$

where $M$ is a positive-definite mass matrix. It is factored as $M^{-1} = B^T B$, with the constraint $\text{Ker}(B) = {0}$.

By default, $M = \mathbf{1}$ (identity) — the base implementation of Apply_B_HMC in the code is a no-op.

A non-trivial mass matrix can precondition the dynamics:

• Assign larger mass to fast modes → they evolve slowly in HMC time.
• Assign smaller mass to slow modes → they evolve faster.
• This can significantly reduce autocorrelation times.

To use a custom mass matrix, override the Apply_B_HMC subroutine in your Hamiltonian submodule. It receives a force/momentum array and a logical flag ltrans:

• ltrans = .false.: apply $B$ (used in momentum updates)
• ltrans = .true.: apply $B^T$ (used in position updates)

Note: No shipped Hamiltonians currently override Apply_B_HMC. This is an advanced feature for users who understand the mode structure of their model.

Validation

The documentation provides a validation benchmark: 6-site Hubbard chain at $\beta t = 4$:

Method	$\langle H \rangle$
HMC only	$-2.81715 \pm 0.00023$
Discrete sequential	$-2.81706 \pm 0.00013$

Use small exact-diagonalization benchmarks like this to validate your HMC setup before production runs.

Known Pitfalls

• HMC requires continuous fields. If any interaction vertex has OP_V%type ≠ 3, the code will terminate with an error. Set Continuous = .true. in your Hamiltonian namelist (e.g. &VAR_Hubbard).
• Forgetting to enable HMC. Setting Delta_t_Langevin_HMC and Leapfrog_Steps without HMC = .true. has no effect.
• Zero defaults. Both Delta_t_Langevin_HMC and Leapfrog_Steps default to 0 — you must set both to nonzero values.
• Langevin vs. HMC. The Delta_t_Langevin_HMC parameter is shared between Langevin and HMC dynamics. Do not enable both simultaneously unless you understand the interplay.